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Let P(3,3) be a point on the hyperbola, x2a2−y2b2=1. If the normal to it at P intersects the x−axis at (9,0) and e is its eccentricity, then the ordered pair (a2,e2) is equal to |
Answer» <p>Let <span class="mjx-chtml" id="MathJax-Element-1-Frame"><span class="mjx-math" id="MJXc-Node-349014"><span class="mjx-mrow" id="MJXc-Node-349015"><span class="mjx-mi" id="MJXc-Node-349016"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.109em;">P</span></span><span class="mjx-mo" id="MJXc-Node-349017"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn" id="MJXc-Node-349018"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mo" id="MJXc-Node-349019"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mn MJXc-space1" id="MJXc-Node-349020"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mo" id="MJXc-Node-349021"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span> be a point on the hyperbola, <span class="mjx-chtml" id="MathJax-Element-2-Frame"><span class="mjx-math" id="MJXc-Node-349022"><span class="mjx-mrow" id="MJXc-Node-349023"><span class="mjx-mstyle" id="MJXc-Node-349024"><span class="mjx-mrow" id="MJXc-Node-349025"><span class="mjx-mfrac" id="MJXc-Node-349026"><span class="mjx-box MJXc-stacked" style="width: 1.161em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.161em; top: -1.528em;"><span class="mjx-msubsup" id="MJXc-Node-349027"><span class="mjx-base"><span class="mjx-mi" id="MJXc-Node-349028"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349029" style=""><span class="mjx-mrow" id="MJXc-Node-349030"><span class="mjx-mn" id="MJXc-Node-349031"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.161em; bottom: -0.853em;"><span class="mjx-msubsup" id="MJXc-Node-349032"><span class="mjx-base"><span class="mjx-mi" id="MJXc-Node-349033"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349034" style=""><span class="mjx-mrow" id="MJXc-Node-349035"><span class="mjx-mn" id="MJXc-Node-349036"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.296em; width: 1.161em;"></span></span><span class="mjx-vsize" style="height: 2.381em; vertical-align: -0.853em;"></span></span></span></span><span class="mjx-mo MJXc-space2" id="MJXc-Node-349037"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mstyle MJXc-space2" id="MJXc-Node-349038"><span class="mjx-mrow" id="MJXc-Node-349039"><span class="mjx-mfrac" id="MJXc-Node-349040"><span class="mjx-box MJXc-stacked" style="width: 1.137em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.137em; top: -1.622em;"><span class="mjx-msubsup" id="MJXc-Node-349041"><span class="mjx-base" style="margin-right: -0.006em;"><span class="mjx-mi" id="MJXc-Node-349042"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.082em; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349043" style=""><span class="mjx-mrow" id="MJXc-Node-349044"><span class="mjx-mn" id="MJXc-Node-349045"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.137em; bottom: -0.854em;"><span class="mjx-msubsup" id="MJXc-Node-349046"><span class="mjx-base"><span class="mjx-mi" id="MJXc-Node-349047"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349048" style=""><span class="mjx-mrow" id="MJXc-Node-349049"><span class="mjx-mn" id="MJXc-Node-349050"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.296em; width: 1.137em;"></span></span><span class="mjx-vsize" style="height: 2.475em; vertical-align: -0.854em;"></span></span></span></span><span class="mjx-mo MJXc-space3" id="MJXc-Node-349051"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3" id="MJXc-Node-349052"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span>. If the normal to it at <span class="mjx-chtml" id="MathJax-Element-3-Frame"><span class="mjx-math" id="MJXc-Node-349053"><span class="mjx-mrow" id="MJXc-Node-349054"><span class="mjx-mi" id="MJXc-Node-349055"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.109em;">P</span></span></span></span></span> intersects the <span class="mjx-chtml" id="MathJax-Element-4-Frame"><span class="mjx-math" id="MJXc-Node-349056"><span class="mjx-mrow" id="MJXc-Node-349057"><span class="mjx-mi" id="MJXc-Node-349058"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo" id="MJXc-Node-349059"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span></span></span></span>axis at <span class="mjx-chtml" id="MathJax-Element-5-Frame"><span class="mjx-math" id="MJXc-Node-349060"><span class="mjx-mrow" id="MJXc-Node-349061"><span class="mjx-mo" id="MJXc-Node-349062"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn" id="MJXc-Node-349063"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">9</span></span><span class="mjx-mo" id="MJXc-Node-349064"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mn MJXc-space1" id="MJXc-Node-349065"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-mo" id="MJXc-Node-349066"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span> and <span class="mjx-chtml" id="MathJax-Element-6-Frame"><span class="mjx-math" id="MJXc-Node-349067"><span class="mjx-mrow" id="MJXc-Node-349068"><span class="mjx-mi" id="MJXc-Node-349069"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span></span></span> is its eccentricity, then the ordered pair <span class="mjx-chtml" id="MathJax-Element-7-Frame"><span class="mjx-math" id="MJXc-Node-349070"><span class="mjx-mrow" id="MJXc-Node-349071"><span class="mjx-mo" id="MJXc-Node-349072"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup" id="MJXc-Node-349073"><span class="mjx-base"><span class="mjx-mi" id="MJXc-Node-349074"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349075" style=""><span class="mjx-mrow" id="MJXc-Node-349076"><span class="mjx-mn" id="MJXc-Node-349077"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span class="mjx-mo" id="MJXc-Node-349078"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-msubsup MJXc-space1" id="MJXc-Node-349079"><span class="mjx-base"><span class="mjx-mi" id="MJXc-Node-349080"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" id="MJXc-Node-349081" style=""><span class="mjx-mrow" id="MJXc-Node-349082"><span class="mjx-mn" id="MJXc-Node-349083"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span class="mjx-mo" id="MJXc-Node-349084"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span> is equal to<br></p> | |