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1.

Consider the circle `x^2 + y^2 = 9` and the parabola `y^2 = 8x`. They intersect at P and Q in first and 4th quadrant,respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.A. 4B. 3C. `8/3`D. 2

Answer» Correct Answer - D
`"Radius of incircle is, r"=Delta/s`
`"Since, "Delta=16sqrt2`
`"Now, s"=(6sqrt2+6sqrt2+4sqrt2)/2`
`=8sqrt2`
`:." "r=(16sqrt2)/(8sqrt2)`
`=2`
Discussion
2.

In triangle `A B C ,a : b : c=4:5: 6.`The ratio of the radius of the circumcircle to that of the incircleis____.

Answer» Correct Answer - `(16)/(7)`
We, have, `R=(abc)/(4Delta)" and "r=Delta/s`
`=(abc)/(4(s-a)(s-b)(s-c))`
But `a : b : c=4 : 5 : 6" "["given"]`
`rArr" "a/4=b/5=c/6=k" [let]"`
`rArr" "a=4k, b=5k, c=6k`
`"Now, "s=1/2(a+b+c)=1/2(4k+5k+6x)=(15k)/2`
`:." "R/r=((4k)(5k)(6k))/(4((15k)/2-4k)((15k)/2-5k)((15k)/2-6k))`
`=(30k^(3))/(k^(3)((15-8)/2)((15-10)/2)((15-12)/2))=(30.8)/(7*5*3)=16/7`
Discussion
3.

In `Delta ABC`, which one is true among the following ?A. `(b+c)"cos"(A)/(2)=sin((B+C)/(2))`B. `(b+c)"cos"((B+C)/(2))=alpha"sin"(A)/(2)`C. `(b-c)"cos"((B-C)/(2))=alpha"cos"((A)/(2))`D. `(b-c)"cos"(A)/(2)=sin((B-C)/(2))`

Answer» Correct Answer - D
Let a,b,c are the sides of `DeltaABC`,
Now, `(b+c)/(c)=(k (sin B+sin C))/(k sin A)` [ by sine rule]
`=(2sin ((B+C)/(2))cos((B-C)/(2)))/(2"sin"(A)/(2)"cos"(A)/(2))rArr(b+c)/(a)=(cos ((B-C)/(2)))/("sin"(A)/(2))`
Also, `(b-c)/(a)=(sin ((B-C)/(2)))/("cos"(A)/(2))`
Discussion
4.

If the angle `A ,Ba n dC`of a triangle are in an arithmetic propression and if `a , ba n dc`denote the lengths of the sides opposite to `A ,Ba n dC`respectively, then the value of the expression `a/csin2C+c/asin2A`is`1/2`(b) `(sqrt(3))/2`(c) `1`(d) `sqrt(3)`A. `(1)/(2)`B. `(sqrt(3))/(2)`C. 1D. `sqrt(3)`

Answer» Correct Answer - D
Since A,B,C are in AP
`rArr 2B=A+C i.e., angle B =60^(@)`
`:. (a)/(c)(2 sin C cos C)+(c)/(a)(2 sin A cos A)`
`=2k(a cos C+c cos A)`
[ Using `(a)/(ainA)=(b)/(sinB)=(c)/(sinC)=(c)/(sin C)=(1)/(k)]`
`=2k(b)`
`=2 sin B " " [ "using" b=a cos C+c cos A]`
`=sqrt(3)`
Discussion