A) \[y=3,y=x+5\]
B) \[y=3,x=3\]
C) \[x=2,\text{ }y=x+5\]
D) \[y=3,\text{ }y=-x+5\]
Correct Answer: D
Solution :
The equation of ellipse is \[9{{x}^{2}}+16{{y}^{2}}=144\] \[\Rightarrow \] \[\frac{{{x}^{2}}}{{{4}^{2}}}+\frac{{{y}^{2}}}{{{3}^{2}}}=1\] ?. (i) This is of the form \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] \[\therefore \] \[{{a}^{2}}={{4}^{2}}\]and \[{{b}^{2}}={{3}^{2}}\] The equation of any tangent to the ellipse (i) is \[y=mx\pm \sqrt{{{a}^{2}}{{m}^{2}}+{{b}^{2}}}\] \[\Rightarrow \] \[y=mx\pm \sqrt{16{{m}^{2}}+9}\] ...(ii) Since, it passes through (2, 3), therefore \[3=2m\pm \sqrt{16{{m}^{2}}+9}\] \[\Rightarrow \] \[{{(3-2m)}^{2}}=(\pm \sqrt{16{{m}^{2}}+9}){{)}^{2}}\] \[\Rightarrow \] \[9+4{{m}^{2}}-12m=16{{m}^{2}}+9\] \[\Rightarrow \] \[12{{m}^{2}}+12m=0\] \[\Rightarrow \] \[{{m}^{2}}+m=0\] \[\Rightarrow \] \[m(m+1)=0\] \[\Rightarrow \] \[m=0,-1\] Substituting these values of\[m\]in Eq. (ii), we get \[y=3\] and \[y=-x+5\]which are the required equation of tangents.
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