The internal common tangents to two circles with centre `C_1` and `C_2` intersect the line joining `C_1` and` C_2` at P and the two direct common tangents intersects the line joining `C_1` and `C_2` at Q. The length `C_1P`, `C_1C_2` are `C_1` Q are in

The internal common tangents to two circles with centre C_1 and C_2 intersect the line joining C_1 and C_2 at P and the two direct common tangents intersects the line joining C_1 and C_2 at Q. The length C_1P , C_1C_2 and C_1 Q are in

The tangent to the curve y=e^x drawn at the point (c,e^c) intersects the line joining the points (c-1,e^(c-1)) and (c+1,e^(c+1))

In the figure given, two circles with centres C_t and C_2 are 35 units apart, i.e. C_1 C_2=35 . The radii of the circles with centres C_1 and C_2 are 12 and 9 respectively. If P is the intersection of C_1 C_2 and a common internal tangent to the circles, then l(C_1 P) equals

Circles C_1 and C_2 are externally tangent and they are both internally tangent to the circle C_3. The radii of C_1 and C_2 are 4 and 10, respectively and the centres of the three circles are collinear. A chord of C_3 is also a common internal tangent of C_1 and C_2. Given that the length of the chord is (msqrtn)/p where m,n and p are positive integers, m and p are relatively prime and n is not divisible by the square of any prime, find the value of (m + n + p).

In the given figure (not to scale), two circles C_1 and C_2 intersect at S and Q. PQN and RQM are tangents drawn to C_1 and C_2 respectively at Q. MAB and ABN are the chords of the circles C_1 and C_2 . If angle NQR =85^@ , then find angle AQB .

The tangent to the curve y=e^(x) drawn at the point (c, e^(c)) intersects the line joining the points (c-1, e^(c-1)) and (c+1, e^(c+1)) .